You can do it either way.
During a haul, the line hand moves down and then back up. While moving down the line hand will pull line down through the guides. When the line hand goes back up to the rod, the line that was pulled out of the guides needs to be fed back into to guides. The up motion may need to be slower than the down motion to allow the line to feed into the rod. If the line hand moves up faster than the line shoots back through the rod, you can introduce slack which can cause your cast may fall apart.
Note: the line hand moves in a continuous down up motion with NO pause between the direction change. Some instructors use the term "downup" as one word to emphasize the continuous motion.
During the up motion, you can release and shoot additional line into the cast, but you don't have to do that. (But, I usually do.
The double haul is all about timing. Start slow with short distance casts to learn the timing and technique before trying to add distance.